From the other hand the fluctuation-dissipation theorem (see, for example the classical textbook [1]}, or its more
advanced version in [2]}) states that the temperature dependent dispersion is given by
\({\bar{\delta }_n}^2(T) = {\bar{\delta }_n^2(0)} \coth \frac{\varepsilon _q}{k_B T}\, .\) 
